After a night of revelry, you find yourself at a park, besides three logical fiends (who are oddly completely naked). Their names are JungleFire, NaturalAntenna, and MorbidThumb.
The following conversation ensues:
JungleFire: "I just picked two integers greater than one."
"Nat, their sum is ..." (he whispers it to NaturalAntenna).
"Morbs, their product is..." (he whispers it to MorbidThumb).
NaturalAntenna: "MorbidThumb, we don't know the numbers."
MorbidThumb: "Now I do.. MuAHahHAHAHAHA..."
Natural Antenna: "As do I... MUAhhAhaHAHAHAHAAa..."
They all look towards you quite condescendingly, and ask in complete unison.. "WHAT ARE THE NUMBERS?"
well, for nat to not know the numbers based on knowing the sum, the sum has to be able to be acheived more than one way. this applies to morbs as well, where the product has to be able to be acheived more than one way. now that nat and morbs know that the other person can't know which pair of integers are correct, they can make an accurate guess at what the numbers are. example: morbs is told that the product is 12, and nat is told the sum is 7. morbs knows it can either be 6x2, or 3x4. morb also knows that nat can't tell which numbers are correct, meaning that there must be at least two different pairs of numbers that equal the sum he was told. his sum could be 8, where nat would be unsure if it is 4+4 or 6+2, or his sum could be 7, where nat would be unsure if it is 3+4 or 5+2. so morbs knows that the numbers could either be 6 and 2, or 3 and 4.
now, nat knows it can either be 3+4, or 5+2. nat also know that morbs can't tell which numbers are correcy, meaning that there must be at least two differnt pairs of numbers that equal the product he was told. his product could be 10, but since the only two numbers whose product is 10 are 5x2, nat knows morbs' number can't be 10. therefore, nat knows morbs' number is 12. now that morbs knows that nat knows, morbs can eliminate 8 as a possiblity as nat's sum, because nat would not be able to know if it was 4+4 or 6+2, but nat would know if it is 5+2 or 3+4, because of the forementioned logic. now that morbs knows that nat's sum is 7, and that the product is 12, morbs can be sure the numbers are 3 and 4. now that nat knows that morbs knows, nat can also be sure that the numbers are 3 and 4.
—le_battement on 2003-06-22 10:54:05
the numbers r.......
2 and 2 since
2+2=4
and 2 x 2=4.
I WON!!!!!!!!!!!!!!!!!!!!!!!!!!!
—dean on 2003-06-22 11:28:40
re: le_battement
If the sum were 7, NaturalAntenna must consider both (2,5) and (3,4). Therefore, since he must consider both pairs, he cannot make the statement 'MorbidThumb, we don't know the numbers.' Making such a statement eliminates the (2,5) pair, since given a product of 10, the only pair of 1+ integers is (2,5).
re: dean
If the sum or product were 4, both logic fiends would immediately know what the numbers were. Thus, NaturalAntenna, as a logic fiend, would not have made the statement 'MorbidThumb, we don't know the numbers.'
—thelogicfiend on 2003-06-22 12:43:47
i guess i don't see what's wrong with eliminating the (2,5) pair. he did make the statement, as seen in the puzzle itself, so if it cancels out that pair in doing so, then what's the problem? =P
—le_battement on 2003-06-22 12:54:36
re: le battement
The (2,5) pair should be eliminated after MorbidThumb's statement. However, if NaturalAntenna said 'MorbidThumb, we don't know the numbers' before Morb's reply, this implies that NaturalAntenna has already eliminated that (2,5). This is because (2,5) contradicts the fact that 'We don't know the numbers.'
—thelogicfiend on 2003-06-22 01:38:01
in other words...
If Nat said 'We don't know the numbers,' that means the sum can't be 7, because then the product might be 10, which means the Morbs already knows.
—thelogicfiend on 2003-06-22 01:40:10
ah, i understand.
—le_battement on 2003-06-22 02:13:31
I don't get it, tell me the answer when someone figures it out.
—Heather on 2003-06-23 02:38:15
I have no fucking clue. Someone tell me please.
—Heather on 2003-06-23 11:29:52
2,6
The numbers are two and six.
Nat recognizes that the only two possible ways to make a sum of eight are 2+6 and 4+4, since 3+5 would mean that Morbs would know the two numbers (since a product of 15 would preclude everything but 3 and 5).
When Morbs gets his product of 12, he knows that it can either be 2 and 6 or 3 and 4.
Thus, neither can definitively solve the problem at this point.
Then , when Nat says that neither of them know the two numbers, Morbs knows that the sum cannot be seven, as 3+4 would give. This is because there are two ways to sum to seven, 2+5 and 3+4. Since if it was 2 and 5 Morbs would know, and if it were 3 and 4 Nat would know if Morbs knew, Morbs knows that the sum cannot be seven, and so the pair of numbers must be 2 and 6.
—Wiggum1006 on 2003-06-24 12:16:50
hmm..
you guys are good. or have way too much time on your hands.
—Heather on 2003-06-24 01:30:55
re: Wiggum
If Nat were given the sum of 8, the possible pairs are (2,6), (3,5), and (4,4). How can he then rule out (3,5) and conclude that both of them don't know the numbers (remember, it's Nat who makes this statement, not Morbs)? This is the same logic you used to later eliminate (3,4).
—theLogicFiend on 2003-06-24 07:59:30
hmmmmmmm........
i see.
—Wiggum1006 on 2003-06-24 08:15:17
2,9
The answer is 2 and 9. Because I say so. I don't want to explain it.
—Wiggum1006 on 2003-06-29 03:26:48
re: Wiggum
Unfortunately, that is not correct.
—theLogicFiend on 2003-07-01 11:50:42
The answer is 3 and 3.
That is what I have concluded.
—Heather on 2003-07-02 02:01:26
I figured It out!!!!!!!
The answer is 2 and 3 because 2+3=5 and the only other thing that = 5 is 4=1 and we can't use 1 cuz the #s are higher than 1. As well 2x3=6 and the only other thing that = 6 is 6x1, but remmeber we can't use 1. Therfore I have come to the conclusion that the integers are 2 & 3.
—Heather on 2003-07-02 02:21:07
re: Heather
Unfortunately, that is also wrong. (2,3) yields a sum of 5 and product of 6, both of which would immediately give away the numbers, considering the fact that they must be integers greater than 1.
—theLogicFiend on 2003-07-02 09:44:41
The answer
6 years too late, but the solution is 4 & 13.
Nat would be told '17', which has possible pairs (2,15),(3,14),(4,13),(5,12),(6,11),(7,10) and (8,9). These give products of 30, 42, 52, 60, 66, 70 and 72. All of these have more than two possible product pairs, so Nat is justified in his initial statement that neither of them know the pair of numbers.
Now Morbs has been told '52', which has possible factor pairs (2,26) and (4,13). Given Nat's statement, he knows that it can't be (2,26), because if Nat had been told '28' then one possible pair is (5,23), whereupon Morbs would have been told '65 and got the answer immediately. So now Morbs knows that the answer must be (4,13).
When Morbs says that he knows the answer, that allows Nat to eliminate all but one of the possible numbers Morbs could have been given.
-If Morbs had been given 30, then he wouldn't have known the pair - it could have been (2,15) or (5,6).
-If Morbs had been given 42, then he wouldn't have known the pair - it could have been (2,21) or (3,14).
-If Morbs had been given 60, then he wouldn't have known the pair - it could have been (3,20) or (5,12).
-If Morbs had been given 66, then he wouldn't have known the pair - it could have been (2,33) or (6,11).
-If Morbs had been given 70, then he wouldn't have known the pair - it could have been (2,35) or (7,10)
-If Morbs had been given 72, then he wouldn't have known the pair - it could have been (3,24) or (8,9).
-However, if Morbs had been given 52, then he would know the pair. This is the only remaining possibility, so Nat can determine that the pair of integers must be (4,13). Ergo, this is the correct answer.
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